Encyclopaedia Britannica > Volume 7, ETM-GOA

3*4
FLUX
~T%-a: From which the pofuion of the point G is
given.
Prob. IV. To f nd the radii of curvature.
The curvature of a circle is uniform in every point,
that of every other curve continually varying : and it
is meafured at any point by that of a circle whofe ra¬
dius is of fuch a length as to coincide with it in cur¬
vature in that point.
All curves that have the fame tangent have the fame
firft fluxion, becaufe the fluxion of a curve and its tan¬
gent are the fame. If it moved uniformly on from
the point of contaft, it would defcribe the tangent.
And the deflexion from the tangent is owing to the
acceleration or retardation of its motion, which is mea¬
fured by its fecond fluxion : and confequently two
curves which have not only the fame tangent, but the
fame curvature at the point of contaft, will have both
their firft and fecond fluxions equal. It is eafily pro¬
ven from thence, that the radius of curvature is
rz t“, x, jy, and z reprefent the abfciifa, ordinate,
— xy
and curve refpeftively.
Examp. Let the given curve be the common para¬
bola, whofe equation isy—a^x* : Then will
A- ‘ . i • ; __
- — x , and (making* conftant) y——^X.^a%x1x T
2 XT
-(wL-v/-?’
/ *3 WflHrH <
the radius of curvature \—.Xy/ 2/fa— "
the vertex, where x—c, will be ~^a.
and
Which at
IONS.
Hence (by the fame rule) the
Fluent of jx’x will be =x3;
8*3
That of 8x*x = ;
3
X-6
That of 2xsx — —;
3
That of
Sometimes the fluent fo found requires to be cor*
refted. The fluxion of x is x, and the fluxion of
is alfo x; becaufe a is invariable, and has therefore no
fluxion.
Now when the fluent of x is required, it muft be
determined, from the nature of the problem,, whether
any invariable part, as a, mud be added to the variable
part x.
When fluents cannot be exa&ly found, they can be
approximated by infinite feries.
Ex. Let it be required to approximate the fluent of
a‘—X”x . . .
~=^rr: m an infinite feries.
'4—* It
The value of —> exprefled in a feries, is ~ 4-
V 1 Vv.4 l -C P
2c3 2ac- ^8rs qac3 8a3cA i6ac
11
i 6rt"V ~\6asc X*6 ^c' Which value being there¬
fore multiplied by xMx, and the fluent taken (by the
common method) we get =—4- — r..x-— 1
S k+iXc 2c3 2uc «+3 +
3 a 1 1 x”-*-5
8r5 qar3 8a3c X»-f 5“^
5 a 3 1 1 *"+7
UkT~ 16ac5 ~ T6^?~ X ^+7 + &i;*
INVERSE METHOD.
From a given fluxion to fnd a fuent.
This is done by tracing back the fteps of the direft
method. The fluxion of x is x ; and therefore the flu¬
ent of x is x : but as there is no direft method of find¬
ing fluents, this branch of the art is imperfeft. We
can affign the fluxion of every fluent; but we cannot
affign the fluent of a fluxion, unlefs it be fuch a one
as may be produced by feme rule in the direft method
from a known fluent.
General Rule, pivide by the fluxion of the root,
add unity to the exponent of the power, and divide by
tire exponent fo increafed.
For,'dividing the fluxion «x—n,x by x (the fluxion
of the root x) it becomes nx”—1; and, adding 1 to the
exponent («—1), we have kx”v which, divided by
gives x”, the true fluent of nxn—'sc. 4
Pros. i. To fnd the area of any curve.
Rule. Multiply the ordinate by the fluxion of the
abfeifla, and the produft gives the fluxion of the figure,
whofe fluent is the area of the figure.
Examp. i. Fig. 8. Let the curve ARMH, whofe
area you will find, be the common parabola. Let u
reprefent the area, and u its fluxion.
In which cafe the relation of AB (x)and BR(y)be-
ing expreifedbyy" — ax (where a is the parameter) we
thence gety = a~xT; and therefore z/= RwiHB (= jx):
zz.aAsfx,: whence arr^-Xa1*3 =:fa^x1 Xx=yj’x (be¬
caufe nLA =y) =ryXABxBR : hence a parabola is
of a reftangle of the fame bafe and altitude.
Examp. 2. Let thepropofed curve CSDR (fig-9.)
be of fuch a nature, that (fuppofing AB unity) the fum
of the areas CSTBC and CDGBC anfwering to any
two propofed abfeiffas AT and AG, ftiall be equal to
the area CRNBC, whofe correfponding abfeifla AN is
equal